y = (8-x) ln x
find the values of x when y = 0

To find the values of \( x \) when \( y = 0 \) for the equation

\[
y = (8 - x) \ln{x}
\]

we need to solve the equation \( (8 - x) \ln{x} = 0 \).

### Step 1: Break the equation into factors
\[
(8 - x) \ln{x} = 0
\]
This equation will be zero if either factor is zero. Thus, we have two cases to consider.

### Case 1: \( \ln{x} = 0 \)
The natural logarithm of \( x \) is zero when \( x = 1 \), since \( \ln{1} = 0 \).

So, one solution is:
\[
x = 1
\]

### Case 2: \( 8 - x = 0 \)
Solve for \( x \):
\[
x = 8
\]

### Step 2: Combine the solutions
The solutions are \( x = 1 \) and \( x = 8 \).

### Final Answer:
The values of \( x \) when \( y = 0 \) are:
\[
x = 1 \quad \text{and} \quad x = 8
\]

Would you like a detailed explanation of how these steps were performed?

Here are some related questions to extend your understanding:

1. What happens to the function as \( x \) approaches 0 from the right?
2. Can \( y = (8 - x) \ln x \) have negative solutions?
3. What is the domain of the function \( y = (8 - x) \ln x \)?
4. What is the behavior of the function as \( x \to \infty \)?
5. How does the graph of \( y = (8 - x) \ln x \) look?

**Tip:** For any logarithmic function, the domain is limited to positive values of \( x \), since \( \ln{x} \) is undefined for \( x \leq 0 \).

y = (8 - x) ln x, find the values of x when y = 0

Learn how to solve the equation y = (8 - x) ln x by finding the values of x when y = 0. This problem involves logarithmic functions, equation solving, and factoring, and provides detailed steps for solving. Suitable for students in Grades 10-12.

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